Upload
others
View
12
Download
0
Embed Size (px)
Citation preview
D. B. Kaplan BAPTS 3/15/19
Entanglement & Symmetry
1
Entanglement by Nuclear Forces and Emergent Symmetries
HEP Seminar, Toronto, March 1, 2019
Martin J Savage
Image owner unknown - from Google Search
Silas Beane David Kaplan Natalie Klco + MJS
arXiv: 1812.03138, to appear in PRL
S. Beane, DBK, N. Klco, M. Savage, PRL 122, 102001 (2019), arXiv: 1812.03138
D. B. Kaplan BAPTS 3/15/19
Symmetry is the theorist’s friend:
D. B. Kaplan BAPTS 3/15/19
Symmetry is the theorist’s friend:
• Gauge symmetries forces
D. B. Kaplan BAPTS 3/15/19
Symmetry is the theorist’s friend:
• Gauge symmetries forces
• (Approximate) global symmetries spectra UV protection suppression of FCNC
D. B. Kaplan BAPTS 3/15/19
Symmetry is the theorist’s friend:
• Gauge symmetries forces
• (Approximate) global symmetries spectra UV protection suppression of FCNC
• Anomalous symmetries η’ mass pion decay axions chiral magnetic effect
D. B. Kaplan BAPTS 3/15/19
Symmetry is the theorist’s friend:
• Gauge symmetries forces
• (Approximate) global symmetries spectra UV protection suppression of FCNC
• Anomalous symmetries η’ mass pion decay axions chiral magnetic effect
D. B. Kaplan BAPTS 3/15/19
Symmetry is the theorist’s friend:
• UV more symmetric than IR ✦ Spontaneously broken symmetries
Weak forces light pions axions
✦ UV fixed point
• Gauge symmetries forces
• (Approximate) global symmetries spectra UV protection suppression of FCNC
• Anomalous symmetries η’ mass pion decay axions chiral magnetic effect
D. B. Kaplan BAPTS 3/15/19
Symmetry is the theorist’s friend:
• UV more symmetric than IR ✦ Spontaneously broken symmetries
Weak forces light pions axions
✦ UV fixed point
• IR more symmetric than UV ✦ IR fixed point ✦ Accidental symmetries
protection from radioactive decay Lorentz symmetry from the lattice
• Gauge symmetries forces
• (Approximate) global symmetries spectra UV protection suppression of FCNC
• Anomalous symmetries η’ mass pion decay axions chiral magnetic effect
D. B. Kaplan BAPTS 3/15/19
There are times when a theorist wishes for more symmetry:• hierarchy • mass textures • suppression of FCNC
Are there other ways that approximate symmetries can emerge?
• “Schrödinger symmetry” (nonrelativistic conformal symmetry) • spin-flavor symmetries
Some evidence in low energy hadronic physics:
This talk: these approximate symmetries can be correlated with minimization of entanglement in scattering.
D. B. Kaplan BAPTS 3/15/19
Emergent symmetries seen in the baryons:
i. SU(4), SU(6) spin-flavor symmetry
ii. SU(4) Wigner symmetry
iii. Schrödinger symmetry
iv. SU(16) (?!) in baryon octet
D. B. Kaplan BAPTS 3/15/19
SU(4), SU(6) spin-flavor symmetry (1960s)
SU(4):
4 =
0
BB@
u "u #d "d #
1
CCA6 =
0
BBBBBB@
u "u #d "d #s "s #
1
CCCCCCA
SU(6):
• Symmetry of non-relativistic quark model • Approximate symmetry apparent in nature:
• masses • magnetic moments & transitions • semi-leptonic currents • meson-baryon couplings • NN scattering
D. B. Kaplan BAPTS 3/15/19
SU(4), SU(6) spin-flavor symmetry (1960s)
• Cannot be a symmetry of relativistic QFT (Coleman-Mandula) • For baryon-meson couplings, does follow from QCD in large-Nc limit
Gervais, Sakita (1984); Dashen, Manohar (1993), Dashen, Jenkins, Manohar (1994) Volume 315, number 3,4 PHYSICS LETTERS B 7 October 1993
~,. ~ SS " ~ o~ ~
(a) (b) (e) Fig. 1. Diagrams contributing to ~N scattering. (c) is suppressed by 1/A~. relative to (a) and (b).
matrix element in the nucleon can be written as
(N[~y i75rav ' lX) = g N e ( N l X i a l N ) , (1)
where (NIX'~l N) and g are of order one. The coupling constant g has been factored out so that the normaliza- tion o f X ia can be chosen conveniently. X i" is an oper- ator (or 4 × 4 matrix) defined on nucleon states which has a finite large-No limit. The pion-nucleon scatter- ing amplitude for 7ra(q) + N ( k ) --, at , (q,) + N ( k ' ) is (neglecting the third graph which is suppressed by l lU~ 2 )
At2 ~2 [~0 - i - i - ' J ' " ~ XJ~X ;~ lx;~xJb] . . . . q,o A '
(2)
where the amplitude is written in the form of an op- erator acting on nucleon states. Both initial and final nucleons are on-shell, so q0 = q,0. The product of the X's in eq. (2) then sums over the possible spins and isospins of the intermediate nucleon. Since f= ~ ,/N,., the overall amplitude is of order ,N~-, which violates unitarity, and also contradicts the large-N,, counting rules of Witten. Thus large-?v~ QCD with a I = J = 1/2 nucleon multiplet interacting with a pion is an inconsistent field theory. There must be other states that cancel the order N~- amplitude in eq. (2) so that the total amplitude is order one, and consistent with unitarity. One can then generalize X ai to be an oper- ator on this degenerate set of baryon states, with ma- trix elements equal to the corresponding axial current matrix elements. With this generalization, the form ofeq. (2) is unchanged. Thus we obtain the first con- sistency condition for QCD,
[ x i ~ , x jb] = 0, (3)
so that the axial currents are represented by a set of operators X ia that commute in the large-N,- limit. The solution of the constraint is non-trivial, because we also have the commutat ion relations
[ j i , xJb] = i eqkX kb, [ I ~ , X jb] = ieabcX jc,
[ J ' , J J ] = i e o k J k, [ I~ , I b] = i e ,bc l ~,
[Ia, j t] = 0, (4)
since X i~ has spin one and isospin one. It is simple to prove that eqs. (3) and (4) form a non-semisimple Lie Algebra, and have no non-trivial finite dimensional representations [7 ].
The solutions of the consistency condition eq. (3) requires that there be an infinite tower of degener- ate baryon states, and also determines the ratios of the p ion-baryon couplings between the states. We will make one simplifying assumption in this letter; the de- generate baryon spectrum consists only of states I = J = 1/2, 3/2 . . . . . where the sequence can be finite or infinite. (One can show that this is the minimal set of states that satisfies the consistency conditions [7].) The states will be denoted by [J, J3, I3). The reduced matrix element of .¥ai between baryon states can be written as
, , i a / 2 J + 1 ( J ' , m ,(,']A [J,m,(t) = x(J,J')v~-f;~_ 1
;H i IH I (~ a (~ /I
where X ( J , J ' ) is a reduced matrix element. The normalization constant has been chosen so that X ( J , J ' ) = X ( J ' , J ) . Since X ia is a tensor with spin one and isospin one, it can only couple states with AJ = 0, ±1, and the independent reduced ma- trix elements are X ( J , J ) and X ( J , J + 1). Taking the matrix element of eq. (3) between l J, ;r/,(t) and [J', m', ~t'), and inserting a complete set of interme- diate states gives
0 = Z (J ' ' ' ' ia (*1) 777 ,(~1X [J l ,m l , Jlmuq
× (J l ,ml , (~l[XJt ' ]J , m,(~) - (ia ~ j b ) . (6)
We will first show that eq. (6) has a unique solution Consider the case J ' = J + 1, where only intermediate
426
D. B. Kaplan BAPTS 3/15/19
• Large-Nc seems to also work in nuclear physics:
1
1/3
gε gρgω gP fρ
0
gf0 fωga0 ga2 gφ
Fig. 3. The couplings for the Nijmegen NN potential in ref. [9] rescaled by fρ.The values for this ratio predicted by large Nc QCD in eq. (4.3) are indicated bylines, and the shaded regions are the size of the expected O(1/Nc) corrections to theleading result. The five regions in the plot (separated by vertical dashed lines) are(from left to right) the (I, S) = (0, 0), (1, 1), (1, 0), (0, 1), and gφ couplings.
potential do not correspond to single meson exchange at all, and the a2 in the Nijmegen
potential has a Gaussian propagator. The model subsumes such effects as 2-π exchange, ρπ
exchange, etc. within the phenomenological couplings gIS . Only the pseudoscalar meson
couplings are related to the physical meson-nucleon couplings, since the long distance part
of the NN potential is dominated by single meson exchange. Thus the agreement between
fig. 3 and the large-Nc prediction (4.3) contains more than the claim that meson-baryon
couplings obey the It = Jt rule. We take fig. 3 to provide encouraging evidence that our
large-Nc analysis of the NN interaction describes the qualitative features seen in nature.
5. The Central Potential and Wigner Supermultiplet Symmetry
It was suggested in ref. [7] that the approximate Wigner supermultiplet symmetry
observed in light nuclei could be explained by the 1/Nc expansion of QCD. Under the
Wigner symmetry SU(4)W , the four nucleon states p ↑, p ↓, n ↑ and n ↓ transform as the
four dimensional fundamental representation. Note that SU(4)W is distinct from the quark
model SU(4), and that the former cannot be realized as a symmetry at the quark level.
Nevertheless, ref. [7] argued that the 1/Nc expansion explains how SU(4)W symmetry
could emerge as an accidental symmetry in light nuclei. As that work only examined the
central part of the NN potential, it is worth reexamining the argument.
12
DBK, A. Manohar (1996)
• …and implies spin-flavor symmetries in low energy baryon-baryon interactions…
D. B. Kaplan BAPTS 3/15/19
SU(2Nf) spin-flavor symmetries in low energy baryon-baryon interactions also follows from large-Nc DBK, M.J. Savage (1995)
• Nf = 2 (nucleons & Δ)
1. Implications of spin-flavor symmetry in effective nuclear forces
Short distance nuclear forces relevant for low energy processes can be incorporated
into chiral Lagrangians in terms of local operators in a derivative expansion [1,2]. There
are two leading (dimension six) operators involving nucleons alone, given by
L6 = −12CS(N †N)2 − 1
2CT (N †σN)2 (1.1)
where N are isodoublet two-component spinors, and the σ are Pauli matrices. Higher
derivative operators account for the spin-orbit coupling, among other effects1. Including
the ∆ isobars in the theory leads to 18 independent dimension six operators allowed by spin
and isospin symmetry2. In order to discuss hypernuclei, or strangeness in dense matter,
one must consider SU(3) flavor multiplets — there are six independent leading operators
involving the baryon octet alone [3], while including the decuplet inflates the number to 28
independent operators. The number of independent dimension seven interactions is still
much greater.
Clearly, to make headway in a systematic effective field theory analysis of nuclear and
hypernuclear forces, it is desirable to find some simplifying principle. In this letter we
propose that among the baryon interactions, SU(4) spin-flavor symmetry for two flavors,
or SU(6) symmetry for three flavors should be a good approximation. We show how these
symmetries have a vastly simplifying effect on the dimension six interactions described
above, reducing both the 18 N−∆ interactions and the 28 octet-decuplet interactions down
to just two independent operators. We support our allegation that spin-flavor symmetry is
relevant to nuclear forces first by outlining its implications and by giving empirical evidence
in support of SU(4) in nuclei. Then we prove that these symmetries become exact in the
large-N limit of QCD.
1 In low energy nucleon-nucleon scattering the higher derivative terms will be less important
than the leading operator. However, many-body effects in large nuclei can enhance the importance
of subleading operators, such as the spin-orbit interaction.2 It is simplest to count operators in the form (ψ1ψ2)(ψ3ψ4)
†, requiring (ψ1ψ2) and (ψ3ψ4) to
have the same spin and isospin quantum numbers. One finds the above two (NN)(NN)† operators;
zero operators of the form (NN)(N∆)†; two (NN)(∆∆)†, four (N∆)(N∆)†, two (N∆)(∆∆)†,
and eight (∆∆)(∆∆)† operators.
1
Under SU(2f) symmetry the two spin states and f flavors of quarks transforming as
the 2f dimensional defining representation. For N = 3, the lowest lying baryons have the
quantum numbers of three quarks in an S-wave, transforming as a three index symmetric
tensor Ψµνρ under SU(2f). For f = 2, Ψ is the 20-dimensional representation of SU(4),
comprising of the four N and sixteen ∆ spin/isospin states; for f = 3, Ψ is the 56-
dimensional SU(6) representation containing the J = 12 octet and the J = 3
2 decuplet. In
either case one finds that there are only two SU(2f) invariant dimension six operators.
These can be written in terms of the baryon fields Ψ as
L6 = −1
f2π
[
a(Ψ†µνρΨ
µνρ)2 + bΨ†µνσΨµντΨ†
ρδτΨρδσ]
, (1.2)
where fπ = 132 MeV is the pion decay constant.
Eq. (1.2) can be expressed in terms of the more familiar fields by writing each SU(2f)
index µ as a pair of flavor and spin indices (iα) under SU(f) × SU(2)J , and then by
projecting out components with the desired SU(f) × SU(2)J transformation properties.
In this way one finds for two flavors
Ψ(αi)(βj)(γk) = ∆ijkαβγ +
1√18
(
N iαϵjkϵβγ + N j
βϵikϵαγ + Nkγ ϵijϵαβ
)
, (1.3)
and for three flavors
Ψ(αi)(βj)(γk) = T ijkαβγ +
1√18
(
Bim,αϵmjkϵβγ + Bj
m,βϵmkiϵγα + Bkm,γϵmijϵαβ
)
. (1.4)
In the above expressions N , ∆, B, and T are the nucleon, isobar, octet and decuplet fields
respectively. Indices i, j . . . denote flavor indices while α, β . . . are spin indices. ∆ and
T are totally symmetric tensors separately in flavor and spin; B is a traceless matrix in
flavor. The normalization factors of 1/√
18 are fixed so that the baryon number operator
equals (Ψ†µνρΨ
µνρ).
By plugging expressions (1.3), (1.4) into our SU(2f) symmetric Lagrangian (1.2) it
is possible to determine all of the leading short range interactions between the J = 12 and
J = 32 baryons in terms of the two coefficients a and b. We spare the reader all of the
results, and focus on predictions for interactions solely involving the J = 12 baryons. For
2
Under SU(2f) symmetry the two spin states and f flavors of quarks transforming as
the 2f dimensional defining representation. For N = 3, the lowest lying baryons have the
quantum numbers of three quarks in an S-wave, transforming as a three index symmetric
tensor Ψµνρ under SU(2f). For f = 2, Ψ is the 20-dimensional representation of SU(4),
comprising of the four N and sixteen ∆ spin/isospin states; for f = 3, Ψ is the 56-
dimensional SU(6) representation containing the J = 12 octet and the J = 3
2 decuplet. In
either case one finds that there are only two SU(2f) invariant dimension six operators.
These can be written in terms of the baryon fields Ψ as
L6 = −1
f2π
[
a(Ψ†µνρΨ
µνρ)2 + bΨ†µνσΨµντΨ†
ρδτΨρδσ]
, (1.2)
where fπ = 132 MeV is the pion decay constant.
Eq. (1.2) can be expressed in terms of the more familiar fields by writing each SU(2f)
index µ as a pair of flavor and spin indices (iα) under SU(f) × SU(2)J , and then by
projecting out components with the desired SU(f) × SU(2)J transformation properties.
In this way one finds for two flavors
Ψ(αi)(βj)(γk) = ∆ijkαβγ +
1√18
(
N iαϵjkϵβγ + N j
βϵikϵαγ + Nkγ ϵijϵαβ
)
, (1.3)
and for three flavors
Ψ(αi)(βj)(γk) = T ijkαβγ +
1√18
(
Bim,αϵmjkϵβγ + Bj
m,βϵmkiϵγα + Bkm,γϵmijϵαβ
)
. (1.4)
In the above expressions N , ∆, B, and T are the nucleon, isobar, octet and decuplet fields
respectively. Indices i, j . . . denote flavor indices while α, β . . . are spin indices. ∆ and
T are totally symmetric tensors separately in flavor and spin; B is a traceless matrix in
flavor. The normalization factors of 1/√
18 are fixed so that the baryon number operator
equals (Ψ†µνρΨ
µνρ).
By plugging expressions (1.3), (1.4) into our SU(2f) symmetric Lagrangian (1.2) it
is possible to determine all of the leading short range interactions between the J = 12 and
J = 32 baryons in terms of the two coefficients a and b. We spare the reader all of the
results, and focus on predictions for interactions solely involving the J = 12 baryons. For
2
• Nf = 2 (restricted to nucleons)
➢ general Weinberg (1990)
➢ SU(4) prediction
two flavors, the SU(4) symmetry yields predictions for the Weinberg coefficients of eq.
(1.1) in terms of a and b:
CS =2(a − b/27)
f2π
, CT = 0 (two flavors). (1.5)
For three flavors, SU(6) symmetry predicts the six Savage-Wise (SW) coefficients
c1 . . . c6 [3] for the interactions involving four baryon octet fields:
c1 = − 727b
c2 = 19b
c3 = 1081b
c4 = −1481b
c5 = a + 29b
c6 = −19b
(three flavors) . (1.6)
Given that CS = (2c1 + c2 + 2c5 + c6)/f2π and CT = (c2 + c6)/f2
π , the SU(6) prediction
(1.6) contains the SU(4) prediction (1.5). Note that the contributions proportional to b
are quite suppressed. In fact, as we discuss below, in the large-N limit of QCD, both a/f2π
and b/f2π are O(N), while there are O(1) (∼ 30%) violations to the SU(6) predictions.
Similarly one expects ∼ 30% violations due to SU(3) breaking by the strange quark mass.
Thus the contributions proportional to b in the above relations are subleading, and it is
consistent to take b = 0 in eq. (1.6) to leading order in N . Note that with the b terms
negligible, the self-interactions among the J = 12 octet baryons are invariant under an
accidental SU(16) symmetry. For two flavors, the prediction CT = 0 implies an accidental
SU(4) symmetry among the nucleon self-interactions.
2. Evidence for SU(4) symmetry in nuclear physics
Why should one believe that there is an approximate SU(2f) symmetry in the short-
range nuclear forces? We first give empirical evidence for SU(4) symmetry in nuclear
physics; we then show that in the large-N limit of QCD, SU(4) relations are accurate to
order 1/N2 (∼ 10%), while SU(6) relations are accurate to O(1/N) (∼ 30%) and O(ms).
The testable consequence of SU(4) symmetry is the prediction (1.5) that CT = 0,
which implies the existence of an accidental symmetry in low energy nucleon interactions.
This symmetry is Wigner’s “supermultiplet symmetry” [4], which we will denote SU(4)sm;
3
D. B. Kaplan BAPTS 3/15/19
Is this an approximate symmetry of nature?
Predicts equal scattering lengths for 1S0 , 3S1 NN scattering lengths
1S0 scattering length = -23.7 fm ~ 1/8 MeV
3S1 scattering length = + 5.4 fm ~ 1/35 MeV
A ' 4⇡
M
1⇣� 1
a + ipME
⌘
very small for both
D. B. Kaplan BAPTS 3/15/19
Is this an approximate symmetry of nature?
Predicts equal scattering lengths for 1S0 , 3S1 NN scattering lengths
1S0 scattering length = -23.7 fm ~ 1/8 MeV
3S1 scattering length = + 5.4 fm ~ 1/35 MeV
A ' 4⇡
M
1⇣� 1
a + ipME
⌘
very small for both
Better diagnostic: accidental SU(4)Wigner symmetry
1. Implications of spin-flavor symmetry in effective nuclear forces
Short distance nuclear forces relevant for low energy processes can be incorporated
into chiral Lagrangians in terms of local operators in a derivative expansion [1,2]. There
are two leading (dimension six) operators involving nucleons alone, given by
L6 = −12CS(N †N)2 − 1
2CT (N †σN)2 (1.1)
where N are isodoublet two-component spinors, and the σ are Pauli matrices. Higher
derivative operators account for the spin-orbit coupling, among other effects1. Including
the ∆ isobars in the theory leads to 18 independent dimension six operators allowed by spin
and isospin symmetry2. In order to discuss hypernuclei, or strangeness in dense matter,
one must consider SU(3) flavor multiplets — there are six independent leading operators
involving the baryon octet alone [3], while including the decuplet inflates the number to 28
independent operators. The number of independent dimension seven interactions is still
much greater.
Clearly, to make headway in a systematic effective field theory analysis of nuclear and
hypernuclear forces, it is desirable to find some simplifying principle. In this letter we
propose that among the baryon interactions, SU(4) spin-flavor symmetry for two flavors,
or SU(6) symmetry for three flavors should be a good approximation. We show how these
symmetries have a vastly simplifying effect on the dimension six interactions described
above, reducing both the 18 N−∆ interactions and the 28 octet-decuplet interactions down
to just two independent operators. We support our allegation that spin-flavor symmetry is
relevant to nuclear forces first by outlining its implications and by giving empirical evidence
in support of SU(4) in nuclei. Then we prove that these symmetries become exact in the
large-N limit of QCD.
1 In low energy nucleon-nucleon scattering the higher derivative terms will be less important
than the leading operator. However, many-body effects in large nuclei can enhance the importance
of subleading operators, such as the spin-orbit interaction.2 It is simplest to count operators in the form (ψ1ψ2)(ψ3ψ4)
†, requiring (ψ1ψ2) and (ψ3ψ4) to
have the same spin and isospin quantum numbers. One finds the above two (NN)(NN)† operators;
zero operators of the form (NN)(N∆)†; two (NN)(∆∆)†, four (N∆)(N∆)†, two (N∆)(∆∆)†,
and eight (∆∆)(∆∆)† operators.
1
X 4 =
0
BB@
p "p #n "n #
1
CCA
D. B. Kaplan BAPTS 3/15/19
β+β+ Ne18
F18
18O1 ,0+
0 ,1+
0 ,1+ 0 ,1
+
2 ,1+2 ,1+
2 ,1+
[1.23] [1.10]1.04
1.98 3.06 1.89
E (MeV) E (MeV) E (MeV)πJ , I πJ , IπJ , I
Fig. 1. Energy level diagram for 18O , 18F and 18Ne. The n − pmass difference and coulomb effects have been removed. The energiesin square brackets are relative to the 18F ground state. β-decays tovarious states are indicated by arrows.
a probability of 87% [15]; the “missing” 13% is largely due to the spin-orbit interaction.
The β-decay strength of transitions between states of this supermultiplet along with the
strengths of transitions to states outside the supermultiplet are shown in Table 1 [16].
It is seen from the first three entries of Table 1 that the Gamow-Teller transitions (∝
στ+) to states within the SU(4) supermultiplet are of similar strength as the superallowed
transition (∝ τ+), 18Neβ+
−→18F (1.04MeV), despite the fact that they are not related by
isospin. Further, it is clear from the last two entries that the decays to states outside the
supermultiplet are much weaker than those to states within the supermultiplet. To be
quantitative, we give the ratio of matrix elements (corrected for Coulomb interactions, gA,
and final state multiplicities) relative to the superallowed transition as the parameter R in
the last column. Isospin predicts equality between the first two entries; SU(4)sm predicts
equality between the first three entries. Allowing for a 13% effect due to contamination of
6
Example of evidence for SU(4)Wigner : β-decay in A=18 isobars
D. B. Kaplan BAPTS 3/15/19
(1,0) + (0,1) = 6 of SU(4)
β+β+ Ne18
F18
18O1 ,0+
0 ,1+
0 ,1+ 0 ,1
+
2 ,1+2 ,1+
2 ,1+
[1.23] [1.10]1.04
1.98 3.06 1.89
E (MeV) E (MeV) E (MeV)πJ , I πJ , IπJ , I
Fig. 1. Energy level diagram for 18O , 18F and 18Ne. The n − pmass difference and coulomb effects have been removed. The energiesin square brackets are relative to the 18F ground state. β-decays tovarious states are indicated by arrows.
a probability of 87% [15]; the “missing” 13% is largely due to the spin-orbit interaction.
The β-decay strength of transitions between states of this supermultiplet along with the
strengths of transitions to states outside the supermultiplet are shown in Table 1 [16].
It is seen from the first three entries of Table 1 that the Gamow-Teller transitions (∝
στ+) to states within the SU(4) supermultiplet are of similar strength as the superallowed
transition (∝ τ+), 18Neβ+
−→18F (1.04MeV), despite the fact that they are not related by
isospin. Further, it is clear from the last two entries that the decays to states outside the
supermultiplet are much weaker than those to states within the supermultiplet. To be
quantitative, we give the ratio of matrix elements (corrected for Coulomb interactions, gA,
and final state multiplicities) relative to the superallowed transition as the parameter R in
the last column. Isospin predicts equality between the first two entries; SU(4)sm predicts
equality between the first three entries. Allowing for a 13% effect due to contamination of
6
Example of evidence for SU(4)Wigner : β-decay in A=18 isobars
D. B. Kaplan BAPTS 3/15/19
(1,0) + (0,1) = 6 of SU(4)
β+β+ Ne18
F18
18O1 ,0+
0 ,1+
0 ,1+ 0 ,1
+
2 ,1+2 ,1+
2 ,1+
[1.23] [1.10]1.04
1.98 3.06 1.89
E (MeV) E (MeV) E (MeV)πJ , I πJ , IπJ , I
Fig. 1. Energy level diagram for 18O , 18F and 18Ne. The n − pmass difference and coulomb effects have been removed. The energiesin square brackets are relative to the 18F ground state. β-decays tovarious states are indicated by arrows.
a probability of 87% [15]; the “missing” 13% is largely due to the spin-orbit interaction.
The β-decay strength of transitions between states of this supermultiplet along with the
strengths of transitions to states outside the supermultiplet are shown in Table 1 [16].
It is seen from the first three entries of Table 1 that the Gamow-Teller transitions (∝
στ+) to states within the SU(4) supermultiplet are of similar strength as the superallowed
transition (∝ τ+), 18Neβ+
−→18F (1.04MeV), despite the fact that they are not related by
isospin. Further, it is clear from the last two entries that the decays to states outside the
supermultiplet are much weaker than those to states within the supermultiplet. To be
quantitative, we give the ratio of matrix elements (corrected for Coulomb interactions, gA,
and final state multiplicities) relative to the superallowed transition as the parameter R in
the last column. Isospin predicts equality between the first two entries; SU(4)sm predicts
equality between the first three entries. Allowing for a 13% effect due to contamination of
6
Gamow-Teller weak transition (β decay): �i⌧+ 2 SU(4)
Example of evidence for SU(4)Wigner : β-decay in A=18 isobars
D. B. Kaplan BAPTS 3/15/19
(1,0) + (0,1) = 6 of SU(4)
SU(4)w allowed transitionsβ+β+ Ne18
F18
18O1 ,0+
0 ,1+
0 ,1+ 0 ,1
+
2 ,1+2 ,1+
2 ,1+
[1.23] [1.10]1.04
1.98 3.06 1.89
E (MeV) E (MeV) E (MeV)πJ , I πJ , IπJ , I
Fig. 1. Energy level diagram for 18O , 18F and 18Ne. The n − pmass difference and coulomb effects have been removed. The energiesin square brackets are relative to the 18F ground state. β-decays tovarious states are indicated by arrows.
a probability of 87% [15]; the “missing” 13% is largely due to the spin-orbit interaction.
The β-decay strength of transitions between states of this supermultiplet along with the
strengths of transitions to states outside the supermultiplet are shown in Table 1 [16].
It is seen from the first three entries of Table 1 that the Gamow-Teller transitions (∝
στ+) to states within the SU(4) supermultiplet are of similar strength as the superallowed
transition (∝ τ+), 18Neβ+
−→18F (1.04MeV), despite the fact that they are not related by
isospin. Further, it is clear from the last two entries that the decays to states outside the
supermultiplet are much weaker than those to states within the supermultiplet. To be
quantitative, we give the ratio of matrix elements (corrected for Coulomb interactions, gA,
and final state multiplicities) relative to the superallowed transition as the parameter R in
the last column. Isospin predicts equality between the first two entries; SU(4)sm predicts
equality between the first three entries. Allowing for a 13% effect due to contamination of
6
Gamow-Teller weak transition (β decay): �i⌧+ 2 SU(4)
Example of evidence for SU(4)Wigner : β-decay in A=18 isobars
D. B. Kaplan BAPTS 3/15/19
SU(4)w disallowed transitions
(1,0) + (0,1) = 6 of SU(4)
SU(4)w allowed transitionsβ+β+ Ne18
F18
18O1 ,0+
0 ,1+
0 ,1+ 0 ,1
+
2 ,1+2 ,1+
2 ,1+
[1.23] [1.10]1.04
1.98 3.06 1.89
E (MeV) E (MeV) E (MeV)πJ , I πJ , IπJ , I
Fig. 1. Energy level diagram for 18O , 18F and 18Ne. The n − pmass difference and coulomb effects have been removed. The energiesin square brackets are relative to the 18F ground state. β-decays tovarious states are indicated by arrows.
a probability of 87% [15]; the “missing” 13% is largely due to the spin-orbit interaction.
The β-decay strength of transitions between states of this supermultiplet along with the
strengths of transitions to states outside the supermultiplet are shown in Table 1 [16].
It is seen from the first three entries of Table 1 that the Gamow-Teller transitions (∝
στ+) to states within the SU(4) supermultiplet are of similar strength as the superallowed
transition (∝ τ+), 18Neβ+
−→18F (1.04MeV), despite the fact that they are not related by
isospin. Further, it is clear from the last two entries that the decays to states outside the
supermultiplet are much weaker than those to states within the supermultiplet. To be
quantitative, we give the ratio of matrix elements (corrected for Coulomb interactions, gA,
and final state multiplicities) relative to the superallowed transition as the parameter R in
the last column. Isospin predicts equality between the first two entries; SU(4)sm predicts
equality between the first three entries. Allowing for a 13% effect due to contamination of
6
Gamow-Teller weak transition (β decay): �i⌧+ 2 SU(4)
Example of evidence for SU(4)Wigner : β-decay in A=18 isobars
D. B. Kaplan BAPTS 3/15/19
SU(4)w disallowed transitions
(1,0) + (0,1) = 6 of SU(4)
SU(4)w allowed transitionsβ+β+ Ne18
F18
18O1 ,0+
0 ,1+
0 ,1+ 0 ,1
+
2 ,1+2 ,1+
2 ,1+
[1.23] [1.10]1.04
1.98 3.06 1.89
E (MeV) E (MeV) E (MeV)πJ , I πJ , IπJ , I
Fig. 1. Energy level diagram for 18O , 18F and 18Ne. The n − pmass difference and coulomb effects have been removed. The energiesin square brackets are relative to the 18F ground state. β-decays tovarious states are indicated by arrows.
a probability of 87% [15]; the “missing” 13% is largely due to the spin-orbit interaction.
The β-decay strength of transitions between states of this supermultiplet along with the
strengths of transitions to states outside the supermultiplet are shown in Table 1 [16].
It is seen from the first three entries of Table 1 that the Gamow-Teller transitions (∝
στ+) to states within the SU(4) supermultiplet are of similar strength as the superallowed
transition (∝ τ+), 18Neβ+
−→18F (1.04MeV), despite the fact that they are not related by
isospin. Further, it is clear from the last two entries that the decays to states outside the
supermultiplet are much weaker than those to states within the supermultiplet. To be
quantitative, we give the ratio of matrix elements (corrected for Coulomb interactions, gA,
and final state multiplicities) relative to the superallowed transition as the parameter R in
the last column. Isospin predicts equality between the first two entries; SU(4)sm predicts
equality between the first three entries. Allowing for a 13% effect due to contamination of
6
Gamow-Teller weak transition (β decay): �i⌧+ 2 SU(4)
SU(4)w allowed matrix elements ~10 x greater than SU(4)w disallowed
Example of evidence for SU(4)Wigner : β-decay in A=18 isobars
D. B. Kaplan BAPTS 3/15/19
So far: no surprises? — emergent spin-flavor symmetries can be explained by large-Nc
D. B. Kaplan BAPTS 3/15/19
So far: no surprises? — emergent spin-flavor symmetries can be explained by large-Nc
First surprise: unnaturally large scattering lengths in NN scattering give approximate Schödinger symmetry (NR conformal)
D. B. Kaplan BAPTS 3/15/19
So far: no surprises? — emergent spin-flavor symmetries can be explained by large-Nc
First surprise: unnaturally large scattering lengths in NN scattering give approximate Schödinger symmetry (NR conformal)
1S0 scattering length = -23.7 fm ~ 1/8 MeV3S1 scattering length = + 5.4 fm ~ 1/35 MeV
A ' 4⇡
M
1⇣� 1
a + ipME
⌘
very small for both
“unitary fermions”,
studied with trapped
atoms
D. B. Kaplan BAPTS 3/15/19
So far: no surprises? — emergent spin-flavor symmetries can be explained by large-Nc
First surprise: unnaturally large scattering lengths in NN scattering give approximate Schödinger symmetry (NR conformal)
1S0 scattering length = -23.7 fm ~ 1/8 MeV3S1 scattering length = + 5.4 fm ~ 1/35 MeV
A ' 4⇡
M
1⇣� 1
a + ipME
⌘
very small for both
“unitary fermions”,
studied with trapped
atoms
Who ordered that??
D. B. Kaplan BAPTS 3/15/19
So far: no surprises? — emergent spin-flavor symmetries can be explained by large-Nc
First surprise: unnaturally large scattering lengths in NN scattering give approximate Schödinger symmetry (NR conformal)
1S0 scattering length = -23.7 fm ~ 1/8 MeV3S1 scattering length = + 5.4 fm ~ 1/35 MeV
A ' 4⇡
M
1⇣� 1
a + ipME
⌘
very small for both
“unitary fermions”,
studied with trapped
atoms
Who ordered that??
D. B. Kaplan BAPTS 3/15/19
So far: no surprises? — emergent spin-flavor symmetries can be explained by large-Nc
First surprise: unnaturally large scattering lengths in NN scattering give approximate Schödinger symmetry (NR conformal)
1S0 scattering length = -23.7 fm ~ 1/8 MeV3S1 scattering length = + 5.4 fm ~ 1/35 MeV
A ' 4⇡
M
1⇣� 1
a + ipME
⌘
very small for both
“unitary fermions”,
studied with trapped
atoms
Who ordered that??
D. B. Kaplan BAPTS 3/15/19
Baryon-baryon interactions Nf=3 / SU(6) spin-flavor symmetry (also predicted by large-Nc)
Under SU(2f) symmetry the two spin states and f flavors of quarks transforming as
the 2f dimensional defining representation. For N = 3, the lowest lying baryons have the
quantum numbers of three quarks in an S-wave, transforming as a three index symmetric
tensor Ψµνρ under SU(2f). For f = 2, Ψ is the 20-dimensional representation of SU(4),
comprising of the four N and sixteen ∆ spin/isospin states; for f = 3, Ψ is the 56-
dimensional SU(6) representation containing the J = 12 octet and the J = 3
2 decuplet. In
either case one finds that there are only two SU(2f) invariant dimension six operators.
These can be written in terms of the baryon fields Ψ as
L6 = −1
f2π
[
a(Ψ†µνρΨ
µνρ)2 + bΨ†µνσΨµντΨ†
ρδτΨρδσ]
, (1.2)
where fπ = 132 MeV is the pion decay constant.
Eq. (1.2) can be expressed in terms of the more familiar fields by writing each SU(2f)
index µ as a pair of flavor and spin indices (iα) under SU(f) × SU(2)J , and then by
projecting out components with the desired SU(f) × SU(2)J transformation properties.
In this way one finds for two flavors
Ψ(αi)(βj)(γk) = ∆ijkαβγ +
1√18
(
N iαϵjkϵβγ + N j
βϵikϵαγ + Nkγ ϵijϵαβ
)
, (1.3)
and for three flavors
Ψ(αi)(βj)(γk) = T ijkαβγ +
1√18
(
Bim,αϵmjkϵβγ + Bj
m,βϵmkiϵγα + Bkm,γϵmijϵαβ
)
. (1.4)
In the above expressions N , ∆, B, and T are the nucleon, isobar, octet and decuplet fields
respectively. Indices i, j . . . denote flavor indices while α, β . . . are spin indices. ∆ and
T are totally symmetric tensors separately in flavor and spin; B is a traceless matrix in
flavor. The normalization factors of 1/√
18 are fixed so that the baryon number operator
equals (Ψ†µνρΨ
µνρ).
By plugging expressions (1.3), (1.4) into our SU(2f) symmetric Lagrangian (1.2) it
is possible to determine all of the leading short range interactions between the J = 12 and
J = 32 baryons in terms of the two coefficients a and b. We spare the reader all of the
results, and focus on predictions for interactions solely involving the J = 12 baryons. For
2
Under SU(2f) symmetry the two spin states and f flavors of quarks transforming as
the 2f dimensional defining representation. For N = 3, the lowest lying baryons have the
quantum numbers of three quarks in an S-wave, transforming as a three index symmetric
tensor Ψµνρ under SU(2f). For f = 2, Ψ is the 20-dimensional representation of SU(4),
comprising of the four N and sixteen ∆ spin/isospin states; for f = 3, Ψ is the 56-
dimensional SU(6) representation containing the J = 12 octet and the J = 3
2 decuplet. In
either case one finds that there are only two SU(2f) invariant dimension six operators.
These can be written in terms of the baryon fields Ψ as
L6 = −1
f2π
[
a(Ψ†µνρΨ
µνρ)2 + bΨ†µνσΨµντΨ†
ρδτΨρδσ]
, (1.2)
where fπ = 132 MeV is the pion decay constant.
Eq. (1.2) can be expressed in terms of the more familiar fields by writing each SU(2f)
index µ as a pair of flavor and spin indices (iα) under SU(f) × SU(2)J , and then by
projecting out components with the desired SU(f) × SU(2)J transformation properties.
In this way one finds for two flavors
Ψ(αi)(βj)(γk) = ∆ijkαβγ +
1√18
(
N iαϵjkϵβγ + N j
βϵikϵαγ + Nkγ ϵijϵαβ
)
, (1.3)
and for three flavors
Ψ(αi)(βj)(γk) = T ijkαβγ +
1√18
(
Bim,αϵmjkϵβγ + Bj
m,βϵmkiϵγα + Bkm,γϵmijϵαβ
)
. (1.4)
In the above expressions N , ∆, B, and T are the nucleon, isobar, octet and decuplet fields
respectively. Indices i, j . . . denote flavor indices while α, β . . . are spin indices. ∆ and
T are totally symmetric tensors separately in flavor and spin; B is a traceless matrix in
flavor. The normalization factors of 1/√
18 are fixed so that the baryon number operator
equals (Ψ†µνρΨ
µνρ).
By plugging expressions (1.3), (1.4) into our SU(2f) symmetric Lagrangian (1.2) it
is possible to determine all of the leading short range interactions between the J = 12 and
J = 32 baryons in terms of the two coefficients a and b. We spare the reader all of the
results, and focus on predictions for interactions solely involving the J = 12 baryons. For
2
baryon decuplet baryon octet
D. B. Kaplan BAPTS 3/15/19
Baryon-baryon interactions Nf=3 / SU(6) spin-flavor symmetry (also predicted by large-Nc)
Under SU(2f) symmetry the two spin states and f flavors of quarks transforming as
the 2f dimensional defining representation. For N = 3, the lowest lying baryons have the
quantum numbers of three quarks in an S-wave, transforming as a three index symmetric
tensor Ψµνρ under SU(2f). For f = 2, Ψ is the 20-dimensional representation of SU(4),
comprising of the four N and sixteen ∆ spin/isospin states; for f = 3, Ψ is the 56-
dimensional SU(6) representation containing the J = 12 octet and the J = 3
2 decuplet. In
either case one finds that there are only two SU(2f) invariant dimension six operators.
These can be written in terms of the baryon fields Ψ as
L6 = −1
f2π
[
a(Ψ†µνρΨ
µνρ)2 + bΨ†µνσΨµντΨ†
ρδτΨρδσ]
, (1.2)
where fπ = 132 MeV is the pion decay constant.
Eq. (1.2) can be expressed in terms of the more familiar fields by writing each SU(2f)
index µ as a pair of flavor and spin indices (iα) under SU(f) × SU(2)J , and then by
projecting out components with the desired SU(f) × SU(2)J transformation properties.
In this way one finds for two flavors
Ψ(αi)(βj)(γk) = ∆ijkαβγ +
1√18
(
N iαϵjkϵβγ + N j
βϵikϵαγ + Nkγ ϵijϵαβ
)
, (1.3)
and for three flavors
Ψ(αi)(βj)(γk) = T ijkαβγ +
1√18
(
Bim,αϵmjkϵβγ + Bj
m,βϵmkiϵγα + Bkm,γϵmijϵαβ
)
. (1.4)
In the above expressions N , ∆, B, and T are the nucleon, isobar, octet and decuplet fields
respectively. Indices i, j . . . denote flavor indices while α, β . . . are spin indices. ∆ and
T are totally symmetric tensors separately in flavor and spin; B is a traceless matrix in
flavor. The normalization factors of 1/√
18 are fixed so that the baryon number operator
equals (Ψ†µνρΨ
µνρ).
By plugging expressions (1.3), (1.4) into our SU(2f) symmetric Lagrangian (1.2) it
is possible to determine all of the leading short range interactions between the J = 12 and
J = 32 baryons in terms of the two coefficients a and b. We spare the reader all of the
results, and focus on predictions for interactions solely involving the J = 12 baryons. For
2
Under SU(2f) symmetry the two spin states and f flavors of quarks transforming as
the 2f dimensional defining representation. For N = 3, the lowest lying baryons have the
quantum numbers of three quarks in an S-wave, transforming as a three index symmetric
tensor Ψµνρ under SU(2f). For f = 2, Ψ is the 20-dimensional representation of SU(4),
comprising of the four N and sixteen ∆ spin/isospin states; for f = 3, Ψ is the 56-
dimensional SU(6) representation containing the J = 12 octet and the J = 3
2 decuplet. In
either case one finds that there are only two SU(2f) invariant dimension six operators.
These can be written in terms of the baryon fields Ψ as
L6 = −1
f2π
[
a(Ψ†µνρΨ
µνρ)2 + bΨ†µνσΨµντΨ†
ρδτΨρδσ]
, (1.2)
where fπ = 132 MeV is the pion decay constant.
Eq. (1.2) can be expressed in terms of the more familiar fields by writing each SU(2f)
index µ as a pair of flavor and spin indices (iα) under SU(f) × SU(2)J , and then by
projecting out components with the desired SU(f) × SU(2)J transformation properties.
In this way one finds for two flavors
Ψ(αi)(βj)(γk) = ∆ijkαβγ +
1√18
(
N iαϵjkϵβγ + N j
βϵikϵαγ + Nkγ ϵijϵαβ
)
, (1.3)
and for three flavors
Ψ(αi)(βj)(γk) = T ijkαβγ +
1√18
(
Bim,αϵmjkϵβγ + Bj
m,βϵmkiϵγα + Bkm,γϵmijϵαβ
)
. (1.4)
In the above expressions N , ∆, B, and T are the nucleon, isobar, octet and decuplet fields
respectively. Indices i, j . . . denote flavor indices while α, β . . . are spin indices. ∆ and
T are totally symmetric tensors separately in flavor and spin; B is a traceless matrix in
flavor. The normalization factors of 1/√
18 are fixed so that the baryon number operator
equals (Ψ†µνρΨ
µνρ).
By plugging expressions (1.3), (1.4) into our SU(2f) symmetric Lagrangian (1.2) it
is possible to determine all of the leading short range interactions between the J = 12 and
J = 32 baryons in terms of the two coefficients a and b. We spare the reader all of the
results, and focus on predictions for interactions solely involving the J = 12 baryons. For
2
baryon decuplet baryon octet
Low-energy EFT for just the octet:
L = �c1TrB†iBiB
†jBj � c2TrB
†iBjB
†jBi � c3TrB
†iB
†jBiBj
�c4TrB†iB
†jBjBi � c5TrB
†iBiTrB
†jBj � c6TrB
†iBjTrB
†jBi
M.J. Savage, M.B. Wise (1995)
Bi =
0
B@
⌃0p2+ ⇤p
6⌃+ p
⌃� �⌃0p2+ ⇤p
6n
⌅� ⌅0 � 2⇤p6
1
CA
i
i = ⬆,⬇
D. B. Kaplan BAPTS 3/15/19
SU(6) prediction:c1 = � 7
27b , c2 =
1
9b , c3 =
10
81b ,
c4 = �14
81b , c5 = a+
2
9b , c6 = �1
9b .
L = �c1TrB†iBiB
†jBj � c2TrB
†iBjB
†jBi � c3TrB
†iB
†jBiBj
�c4TrB†iB
†jBjBi � c5TrB
†iBiTrB
†jBj � c6TrB
†iBjTrB
†jBi
Does this work? Look at lattice data • NPLQCD collaboration, 2015 • equal quark masses • mπ = 806 MeV
D. B. Kaplan BAPTS 3/15/19
30
Case c1 c2 c3 c4 c5 c6
Unnatural 0.051(+53)(�64) 0.073(+64)
(�54) 0.088(+53)(�55) 0.088(+58)
(�55) 1.892(+59)(�50) �0.013(+54)
(�63)
Natural 5(+17)(�12) 7(+16)
(�13) 5(+12)(�8) 5(+11)
(�12) �19(+12)(�17) �4(+14)
(�16)
1
TABLE VI: Values of the coefficients of the LO SU(3)-symmetric interactions obtained by solving Eqs. (29)-(34) for the unnatural case with µ = m⇡, and for the natural case with µ = 0. The coefficients are expressedin units of [ 2⇡
MB], with MB being the baryon mass in this calculation, expressed in lattice units.
in the 8S and 1 irreps, and provides further constraints on the SW coefficients,
(�2c13
+2c23
�5c36
+5c46
+ c5 � c6)�1
� µ = �0.08(4) l.u. (33)
(�c13
+c23
�8c33
+8c43
+ c5 � c6)�1
� µ = �0.08(4) l.u. (34)
Setting µ = 0 recovers the results for natural systems.Eqs. (29)-(34) are solved to determine all six SW coefficients for unnatural interactions within the
KSW-vK power counting at a renormalization scale of µ = m⇡, and for natural interactions througha tree-level expansion of the scattering amplitude, see Table VI. As is evident from these values,shown in Fig. 16, the unnatural scenario provides the most stringent constraints on the coefficients.In this case, the value of all SW coefficients except for c5 are consistent with zero, a manifestationof the SU(16) spin-flavor symmetry in the LO SU(3) interactions, i.e., the a � b/3 hierarchy inthe SU(6) spin-flavor symmetric interactions. With these results, and the binding energies of lighthypernuclei [20], ongoing ab initio many-body calculations using the LQCD input at this value ofthe quark masses [59–62] can be extended to systems containing hyperons. Appendix B is devotedto summarizing the constraints obtained for the LO scattering amplitudes in flavor space.
Natural caseUnnatural case @ µ = m⇡
c1 c2 c3 c4 c5 c6 c1 c2 c3 c4 c5 c6
�� �� �� ��
��
��
-��
-��
-��
�
��
��
��
�� �� �� ��
��
�����
���
���
���
���
FIG. 16: A comparison of the coefficients of the LO SU(3)-symmetric interactions. The left panel corre-sponds to the unnatural case with µ = m⇡, while the right panel represents the natural case with µ = 0,corresponding to a tree-level expansion of the scattering amplitudes. The coefficients are expressed in unitsof [ 2⇡
MB], with MB being the baryon mass in this calculation, expressed in lattice units.
Baryon-baryon interactions and spin-flavor symmetry from latticequantum chromodynamics
Michael L. Wagman,1,2 Frank Winter,3 Emmanuel Chang,2 Zohreh Davoudi,4 William Detmold,4
Kostas Orginos,5,3 Martin J. Savage,1,2 and Phiala E. Shanahan4
(NPLQCD Collaboration)
1Department of Physics, University of Washington, Box 351560, Seattle, Washington 98195, USA2Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195-1550, USA
3Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, Virginia 23606, USA4Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA5Department of Physics, College of William and Mary, Williamsburg, Virginia 23187-8795, USA
(Received 25 July 2017; published 28 December 2017)
Lattice quantum chromodynamics is used to constrain the interactions of two octet baryons at the SUð3Þflavor-symmetric point, with quark masses that are heavier than those in nature (equal to that of the physicalstrange quark mass and corresponding to a pion mass of ≈806 MeV). Specifically, the S-wave scatteringphase shifts of two-baryon systems at low energies are obtained with the application of Lüscher’sformalism, mapping the energy eigenvalues of two interacting baryons in a finite volume to the two-particlescattering amplitudes below the relevant inelastic thresholds. The leading-order low-energy scatteringparameters in the two-nucleon systems that were previously obtained at these quark masses are determinedwith a refined analysis, and the scattering parameters in two other channels containing the Σ and Ξ baryonsare constrained for the first time. It is found that the values of these parameters are consistent with anapproximate SUð6Þ spin-flavor symmetry in the nuclear and hypernuclear forces that is predicted in thelarge-Nc limit of QCD. The two distinct SUð6Þ-invariant interactions between two baryons are constrainedfor the first time at this value of the quark masses, and their values indicate an approximate accidentalSUð16Þ symmetry. The SUð3Þ irreps containing the NNð1S0Þ, NNð3S1Þ and 1ffiffi
2p ðΞ0nþ Ξ−pÞð3S1Þ channels
unambiguously exhibit a single bound state, while the irrep containing the Σþpð3S1Þ channel exhibits astate that is consistent with either a bound state or a scattering state close to threshold. These results are inagreement with the previous conclusions of the NPLQCD collaboration regarding the existence of two-nucleon bound states at this value of the quark masses.
DOI: 10.1103/PhysRevD.96.114510
I. INTRODUCTION
It is speculated that hyperons, the counterparts of nucleonsin which some of the valence quarks in the nucleon arereplaced by strange quarks, play an important role in thecomposition of dense matter, such as that in the interior ofneutron stars (for a comprehensive review, see Ref. [1]). Theinteractions between two nucleons are precisely constrainedby experiment over a wide range of energies. However,those between a nucleon and a hyperon, or between twohyperons, are not well known [2–19], and are challengingto probe experimentally because of the short lifetime ofhyperons and hypernuclei. Precise information on howhyperons interact, in particular in a nuclear medium, isessential to establish their effects on the equation of state ofdense matter and other observables. On the theoretical side,the only reliable method with which to determine theseinteractions is to calculate them from the underlying stronginteractions among quarks and gluons described by quantum
chromodynamics (QCD). This can be achieved using thenonperturbative method of lattice QCD (LQCD), whichinvolves numerically evaluating path integrals representingEuclidean correlation functions using Monte Carlo samplingmethods. This approach is taken in this work to constrain thescattering amplitudes of several classes of nucleon-nucleon,hyperon-nucleon and hyperon-hyperon systems, albeit in aworld that exhibits an exact SUð3Þ flavor symmetry, withdegenerate light and strange quark masses tuned to producepions and kaons with masses of ≈806 MeV. The calcu-lations are performed in the absence of quantum electrody-namics (QED). This work extends our previous studies ofsuch systems using the same ensembles of gauge-fieldconfigurations [20,21], and complements previous andongoing studies of hyperon interactions using LQCD, seefor example Refs. [19,22–35].Lüscher’s finite-volume (FV) methodology [36–51] is
used to constrain the scattering amplitudes of two-baryonsystems below the relevant inelastic thresholds from the
PHYSICAL REVIEW D 96, 114510 (2017)
2470-0010=2017=96(11)=114510(36) 114510-1 © 2017 American Physical Society
INT-PUB-17-017, MIT-CTP-4912, NSF-ITP-17-076
Baryon-Baryon Interactions and Spin-Flavor Symmetryfrom Lattice Quantum Chromodynamics
Michael L. Wagman,1, 2 Frank Winter,3 Emmanuel Chang, Zohreh Davoudi,4
William Detmold,4 Kostas Orginos,5, 3 Martin J. Savage,1, 2 and Phiala E. Shanahan4
(NPLQCD Collaboration)1Department of Physics, University of Washington, Box 351560, Seattle, WA 98195, USA2Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA
3Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606, USA
4Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
5Department of Physics, College of William and Mary, Williamsburg, VA 23187-8795, USA
(Dated: May 17, 2018)
Lattice quantum chromodynamics is used to constrain the interactions of two octet baryonsat the SU(3) flavor-symmetric point, with quark masses that are heavier than those in na-ture (equal to that of the physical strange quark mass and corresponding to a pion mass of⇡ 806 MeV). Specifically, the S-wave scattering phase shifts of two-baryon systems at lowenergies are obtained with the application of Lüscher’s formalism, mapping the energy eigen-values of two interacting baryons in a finite volume to the two-particle scattering amplitudesbelow the relevant inelastic thresholds. The values of the leading-order low-energy scatteringparameters in the irreducible representations of SU(3) are consistent with an approximateSU(6) spin-flavor symmetry in the nuclear and hypernuclear forces that is predicted in thelarge-Nc limit of QCD. The two distinct SU(6)-invariant interactions between two baryonsare constrained at this value of the quark masses, and their values indicate an approximateaccidental SU(16) symmetry. The SU(3) irreps containing the NN (1S0), NN (3S1) and1p2(⌅0n + ⌅�p) (3S1) channels unambiguously exhibit a single bound state, while the irrep
containing the ⌃+p (3S1) channel exhibits a state that is consistent with either a boundstate or a scattering state close to threshold. These results are in agreement with the previ-ous conclusions of the NPLQCD collaboration regarding the existence of two-nucleon boundstates at this value of the quark masses.
PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.-t, 21.30.Fe, 13.75.Cs, 13.85.-t.
I. INTRODUCTION
It is speculated that hyperons, the counterparts of nucleons in which some of the valence quarksin the nucleon are replaced by strange quarks, play an important role in the composition of densematter, such as that in the interior of neutron stars (for a comprehensive review, see Ref. [1]).The interactions between two nucleons are precisely constrained by experiment over a wide rangeof energies. However, those between a nucleon and a hyperon, or between two hyperons, are notwell known [2–19], and are challenging to probe experimentally because of the short lifetime ofhyperons and hypernuclei. Precise information on how hyperons interact, in particular in a nuclearmedium, is essential to establish their effects on the equation of state of dense matter and otherobservables. On the theoretical side, the only reliable method with which to determine theseinteractions is to calculate them from the underlying strong interactions among quarks and gluonsdescribed by quantum chromodynamics (QCD). This can be achieved using the non-perturbativemethod of lattice QCD (LQCD), which involves numerically evaluating path integrals representingEuclidean correlation functions using Monte Carlo sampling methods. This approach is taken
arX
iv:1
706.
0655
0v2
[hep
-lat]
26
Jul 2
017
m⇡ ⇡ 806 MeVµ = m⇡
D. B. Kaplan BAPTS 3/15/19
X X XXX
L = �c1TrB†iBiB
†jBj � c2TrB
†iBjB
†jBi � c3TrB
†iB
†jBiBj
�c4TrB†iB
†jBjBi � c5 TrB
†iBi TrB
†jBj � c6 TrB
†iBj TrB†
jBi
NPLQCD results: • Only c5 ≠ 0 • Near critical value for large scattering lengths
Similar to Nf=2 large-Nc result:
1. Implications of spin-flavor symmetry in effective nuclear forces
Short distance nuclear forces relevant for low energy processes can be incorporated
into chiral Lagrangians in terms of local operators in a derivative expansion [1,2]. There
are two leading (dimension six) operators involving nucleons alone, given by
L6 = −12CS(N †N)2 − 1
2CT (N †σN)2 (1.1)
where N are isodoublet two-component spinors, and the σ are Pauli matrices. Higher
derivative operators account for the spin-orbit coupling, among other effects1. Including
the ∆ isobars in the theory leads to 18 independent dimension six operators allowed by spin
and isospin symmetry2. In order to discuss hypernuclei, or strangeness in dense matter,
one must consider SU(3) flavor multiplets — there are six independent leading operators
involving the baryon octet alone [3], while including the decuplet inflates the number to 28
independent operators. The number of independent dimension seven interactions is still
much greater.
Clearly, to make headway in a systematic effective field theory analysis of nuclear and
hypernuclear forces, it is desirable to find some simplifying principle. In this letter we
propose that among the baryon interactions, SU(4) spin-flavor symmetry for two flavors,
or SU(6) symmetry for three flavors should be a good approximation. We show how these
symmetries have a vastly simplifying effect on the dimension six interactions described
above, reducing both the 18 N−∆ interactions and the 28 octet-decuplet interactions down
to just two independent operators. We support our allegation that spin-flavor symmetry is
relevant to nuclear forces first by outlining its implications and by giving empirical evidence
in support of SU(4) in nuclei. Then we prove that these symmetries become exact in the
large-N limit of QCD.
1 In low energy nucleon-nucleon scattering the higher derivative terms will be less important
than the leading operator. However, many-body effects in large nuclei can enhance the importance
of subleading operators, such as the spin-orbit interaction.2 It is simplest to count operators in the form (ψ1ψ2)(ψ3ψ4)
†, requiring (ψ1ψ2) and (ψ3ψ4) to
have the same spin and isospin quantum numbers. One finds the above two (NN)(NN)† operators;
zero operators of the form (NN)(N∆)†; two (NN)(∆∆)†, four (N∆)(N∆)†, two (N∆)(∆∆)†,
and eight (∆∆)(∆∆)† operators.
1
X
D. B. Kaplan BAPTS 3/15/19
X X XXX
L = �c1TrB†iBiB
†jBj � c2TrB
†iBjB
†jBi � c3TrB
†iB
†jBiBj
�c4TrB†iB
†jBjBi � c5 TrB
†iBi TrB
†jBj � c6 TrB
†iBj TrB†
jBi
NPLQCD results: • Only c5 ≠ 0 • Near critical value for large scattering lengths
Similar to Nf=2 large-Nc result:
1. Implications of spin-flavor symmetry in effective nuclear forces
Short distance nuclear forces relevant for low energy processes can be incorporated
into chiral Lagrangians in terms of local operators in a derivative expansion [1,2]. There
are two leading (dimension six) operators involving nucleons alone, given by
L6 = −12CS(N †N)2 − 1
2CT (N †σN)2 (1.1)
where N are isodoublet two-component spinors, and the σ are Pauli matrices. Higher
derivative operators account for the spin-orbit coupling, among other effects1. Including
the ∆ isobars in the theory leads to 18 independent dimension six operators allowed by spin
and isospin symmetry2. In order to discuss hypernuclei, or strangeness in dense matter,
one must consider SU(3) flavor multiplets — there are six independent leading operators
involving the baryon octet alone [3], while including the decuplet inflates the number to 28
independent operators. The number of independent dimension seven interactions is still
much greater.
Clearly, to make headway in a systematic effective field theory analysis of nuclear and
hypernuclear forces, it is desirable to find some simplifying principle. In this letter we
propose that among the baryon interactions, SU(4) spin-flavor symmetry for two flavors,
or SU(6) symmetry for three flavors should be a good approximation. We show how these
symmetries have a vastly simplifying effect on the dimension six interactions described
above, reducing both the 18 N−∆ interactions and the 28 octet-decuplet interactions down
to just two independent operators. We support our allegation that spin-flavor symmetry is
relevant to nuclear forces first by outlining its implications and by giving empirical evidence
in support of SU(4) in nuclei. Then we prove that these symmetries become exact in the
large-N limit of QCD.
1 In low energy nucleon-nucleon scattering the higher derivative terms will be less important
than the leading operator. However, many-body effects in large nuclei can enhance the importance
of subleading operators, such as the spin-orbit interaction.2 It is simplest to count operators in the form (ψ1ψ2)(ψ3ψ4)
†, requiring (ψ1ψ2) and (ψ3ψ4) to
have the same spin and isospin quantum numbers. One finds the above two (NN)(NN)† operators;
zero operators of the form (NN)(N∆)†; two (NN)(∆∆)†, four (N∆)(N∆)†, two (N∆)(∆∆)†,
and eight (∆∆)(∆∆)† operators.
1
X• More symmetric than SU(6) [b ≈ 0] • EFT possesses SU(16) analog of SU(4)Wigner • Near critical value for large scattering lengths — conformal symmetry
But:
Not large-Nc predictions
D. B. Kaplan BAPTS 3/15/19
Nf=2 Nf=3
SU(4)Wigner SU(16)NPLQCD
~conformal ~conformal
D. B. Kaplan BAPTS 3/15/19
Nf=2 Nf=3
SU(4)Wigner SU(16)NPLQCD
~conformal ~conformal
No known reason for these symmetries
D. B. Kaplan BAPTS 3/15/19
Nf=2 Nf=3
SU(4)Wigner SU(16)NPLQCD
~conformal ~conformal
No known reason for these symmetries
…but correlated with low entanglement
D. B. Kaplan BAPTS 3/15/19
D. B. Kaplan BAPTS 3/15/19
When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives (or wave-functions) have become entangled.
555
DISCUSSION OF PROBABILITY RELATIONS BETWEENSEPARATED SYSTEMS
BY E. SCHRODINGER
[Communicated by Mr M. BORN]
[Received 14 August, read 28 October 1935]
1. When two systems, of which we know the states by their respective repre-sentatives, enter into temporary physical interaction due to knownforces betweenthem, and when after a time of mutual influence the systems separate again, thenthey can no longer be described in the same way as before, viz. by endowing eachof them with a representative of its own. I would not call that one but rather thecharacteristic trait of quantum mechanics," the one that enforces its entiredeparture from classical lines of thought. By the interaction the two repre-sentatives (or ^-functions) have become entangled. To disentangle them we mustgather further information by experiment, although we knew as much as any-body could possibly know about all that happened. Of either system, takenseparately, all previous knowledge may be entirely lost, leaving us but oneprivilege: to restrict the experiments to one only of the two systems. After re-establishing one representative by observation, the other one can be inferredsimultaneously. In what follows the whole of this procedure will be called thedisentanglement. Its sinister importance is due to its being involved in everymeasuring process and therefore forming the basis of the quantum theory ofmeasurement, threatening us thereby with at least a regressus in infinitum, sinceit will be noticed that the procedure itself involves measurement.
Another way of expressing the peculiar situation is: the best possible knowledgeof a whole does not necessarily include the best possible knowledge of all its parts,even though they may be entirely separated and therefore virtually capable ofbeing " best possibly known ", i.e. of possessing, each of them, a representative ofits own. The lack of knowledge is by no means due to the interaction being in-sufficiently known—at least not in the way that it could possibly be known morecompletely—it is due to the interaction itself.
Attention has recently* been called to the obvious but very disconcerting factthat even though we restrict the disentangling measurements to one system, therepresentative obtained for the other system is by no means independent of theparticular choice of observations which we select for that purpose and which by
* A. Einstein, B. Podolsky and N. Rosen, Phya. Rev. 47 (1935), 777.
D7 C C7 3 3 34 7 3D :DD C 53 4 697 9 5 7 D7 C :DD C 6 9 03676 :DD C 53 4 697 9 5 7 1 7 C D 23C: 9D . 4 3 7C /3 3D C 4 75D D D:7 ,3 4 697 , 7
D. B. Kaplan BAPTS 3/15/19
EPRLOCAL
REALITY ASSUMPTION
D. B. Kaplan BAPTS 3/15/19
How to quantify entanglement?
D. B. Kaplan BAPTS 3/15/19
How to quantify entanglement?Quantum entropy:
S = - Tr ρ ln ρ, ρ = density matrix
D. B. Kaplan BAPTS 3/15/19
How to quantify entanglement?Quantum entropy:
S = - Tr ρ ln ρ, ρ = density matrix
E.g. pure state: | i = | "x #y i
⇢ = | ih |
⇢ =
0
B@1 0 . . .0 0 . . ....
.... . .
1
CAS = 0
➤ rank 1
D. B. Kaplan BAPTS 3/15/19
How to quantify entanglement?Quantum entropy:
S = - Tr ρ ln ρ, ρ = density matrix
E.g. pure state: | i = | "x #y i
⇢ = | ih |
⇢ =
0
B@1 0 . . .0 0 . . ....
.... . .
1
CAS = 0
➤ rank 1
E.g. mixed state:
S = ln 2
➤ rank 2⇢ =
1
2
0
BBB@
1 0 0 . . .0 1 0 . . .0 0 0 . . ....
......
. . .
1
CCCA
D. B. Kaplan BAPTS 3/15/19
How to quantify entanglement?Quantum entropy:
S = - Tr ρ ln ρ, ρ = density matrix
E.g. pure state: | i = | "x #y i
⇢ = | ih |
⇢ =
0
B@1 0 . . .0 0 . . ....
.... . .
1
CAS = 0
➤ rank 1
E.g. mixed state:
S = ln 2
➤ rank 2⇢ =
1
2
0
BBB@
1 0 0 . . .0 1 0 . . .0 0 0 . . ....
......
. . .
1
CCCA
D. B. Kaplan BAPTS 3/15/19
H = HA x HB
A
B
Reduced density matrix:
Factorizable Hilbert space:
⇢A = TrB ⇢
⇢B = TrA ⇢
D. B. Kaplan BAPTS 3/15/19
H = HA x HB
A
B
Reduced density matrix:
Factorizable Hilbert space:
⇢A = TrB ⇢
⇢B = TrA ⇢
Pure state on H — typically ρA, ρB will represent mixed states, reflected in entropy:
S = 0 , SA = SB 6= 0
Shows that systems A and B are entangled
D. B. Kaplan BAPTS 3/15/19
Is there a connection between entanglement and dynamics?
D. B. Kaplan BAPTS 3/15/19
Is there a connection between entanglement and dynamics?
Yes: for many-body systems, QFTs: when A, B correspond to spatial regions
D. B. Kaplan BAPTS 3/15/19
Is there a connection between entanglement and dynamics?
Yes: for many-body systems, QFTs: when A, B correspond to spatial regions
|nx1, nx2, nx3, nx4, … ny1, ny2, ny3, … >
A B
D. B. Kaplan BAPTS 3/15/19
Is there a connection between entanglement and dynamics?
Yes: for many-body systems, QFTs: when A, B correspond to spatial regions
|nx1, nx2, nx3, nx4, … ny1, ny2, ny3, … >
A B
Observed that ground states seem to obey area-law entanglement
SA = SB / area of shared boundary
A
B
D. B. Kaplan BAPTS 3/15/19
Is there a connection between entanglement and dynamics?
Yes: for many-body systems, QFTs: when A, B correspond to spatial regions
|nx1, nx2, nx3, nx4, … ny1, ny2, ny3, … >
A B
Observed that ground states seem to obey area-law entanglement
SA = SB / area of shared boundary
A
B
What is special about position coordinates?
Hamiltonian is local Correlations fall off with |x-y|
D. B. Kaplan BAPTS 3/15/19
Entanglement knows about dynamics
In a strongly coupled system with composite particles (eg, QCD) can entanglement help determine their wave functions and interactions (and hence their symmetries)?
Quantify the amount of entanglement in the S-matrix
D. B. Kaplan BAPTS 3/15/19
How to quantify entanglement of a N-N scattering process?
One way (PRL 122, 102001 (2019), arXiv: 1812.03138):
D. B. Kaplan BAPTS 3/15/19
How to quantify entanglement of a N-N scattering process?
One way (PRL 122, 102001 (2019), arXiv: 1812.03138):
• Start with two non-identical baryon flavors with spin up (pure state, zero entropy)
| "i1 | "i2
D. B. Kaplan BAPTS 3/15/19
How to quantify entanglement of a N-N scattering process?
One way (PRL 122, 102001 (2019), arXiv: 1812.03138):
• Start with two non-identical baryon flavors with spin up (pure state, zero entropy)
| "i1 | "i2
• Rotate the two spins independently, this is the in state (pure state, zero entropy)
R(⌦1)| "i1 R(⌦2)| "i2
D. B. Kaplan BAPTS 3/15/19
How to quantify entanglement of a N-N scattering process?
One way (PRL 122, 102001 (2019), arXiv: 1812.03138):
• Start with two non-identical baryon flavors with spin up (pure state, zero entropy)
| "i1 | "i2
• Rotate the two spins independently, this is the in state (pure state, zero entropy)
R(⌦1)| "i1 R(⌦2)| "i2
• Compute the out-state using the S-matrix
D. B. Kaplan BAPTS 3/15/19
How to quantify entanglement of a N-N scattering process?
One way (PRL 122, 102001 (2019), arXiv: 1812.03138):
• Start with two non-identical baryon flavors with spin up (pure state, zero entropy)
| "i1 | "i2
• Rotate the two spins independently, this is the in state (pure state, zero entropy)
R(⌦1)| "i1 R(⌦2)| "i2
• Compute the out-state using the S-matrix
• Compute the reduced density matrix ρ1 for the out-state
D. B. Kaplan BAPTS 3/15/19
How to quantify entanglement of a N-N scattering process?
One way (PRL 122, 102001 (2019), arXiv: 1812.03138):
• Start with two non-identical baryon flavors with spin up (pure state, zero entropy)
| "i1 | "i2
• Rotate the two spins independently, this is the in state (pure state, zero entropy)
R(⌦1)| "i1 R(⌦2)| "i2
• Compute the out-state using the S-matrix
• Compute the reduced density matrix ρ1 for the out-state
• Define the entanglement power for the S-matrix:
INT-PUB-18-056NT@UW-18-19
Entanglement Suppression and Emergent Symmetries of Strong Interactions
Silas R. Beane,1 David B. Kaplan,2 Natalie Klco,1, 2 and Martin J. Savage2
1Department of Physics, University of Washington, Seattle, WA 98195-1560, USA
2Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA
(Dated: December 10, 2018 - 1:30)
Entanglement suppression in the strong interaction S-matrix is shown to be correlated with ap-proximate spin-flavor symmetries that are observed in low-energy baryon interactions, the WignerSU(4) symmetry for two flavors and an SU(16) symmetry for three flavors. We conjecture thatdynamical entanglement suppression is a property of the strong interactions in the infrared, givingrise to these emergent symmetries and providing powerful constraints on the nature of nuclear andhypernuclear forces in dense matter.
Understanding approximate global symmetries in thestrong interactions has played an important historicalrole in the development of the theory of Quantum Chro-modynamics (QCD). Baryon number symmetry arises inQCD because it is impossible to include a marginal orrelevant interaction consistent with Lorentz and gaugesymmetry which violates baryon number, while the ax-ial and vector flavor symmetries are understood to bedue to the small ratio of quark masses (and their di↵er-ences) to the QCD scale. The approximate low-energySU(2nf ) spin-flavor symmetry for nf = 2, 3 flavors whichrelates spin-1/2 and spin-3/2 baryons can be understoodas arising at leading order (LO) in the large-Nc expan-sion, where Nc is the number of colors [1, 2]. In low-energy nuclear physics, a di↵erent spin-flavor symmetryis observed in the structure of light-nuclei and their �-decay rates, namely Wigner’s SU(4) symmetry, wherethe two spin states of the two nucleons transform asthe 4-dimensional fundamental representation [3–5]. Ithas been shown that this symmetry also arises from thelarge-Nc expansion at energies below the � mass [6–8].The agreement of large-Nc predictions with nuclear phe-nomenology has been extended to higher-order interac-tions [9–12], three-nucleon systems [13–15], and to stud-ies of hadronic parity violation [16–18]. Recently, how-ever, lattice QCD computations for nf = 3 have revealedan emergent SU(16) symmetry in low-energy interactionsof the baryon octet—analogous to Wigner’s SU(4), butwith the two spin states of the eight baryons transformingas the 16-dimensional representation of SU(16) [19]. Thislow energy symmetry has been lacking an explanationfrom QCD. In this Letter, we show that both Wigner’sSU(4) symmetry for nf = 2 and SU(16) for nf = 3correspond to fixed lines of minimal quantum entangle-ment in the S-matrix for baryon-baryon scattering, andwe propose entanglement suppression to be a dynami-cal property of QCD and the origin of these emergentsymmetries 1.
1 A principle of maximum entanglement has been previously pro-posed to constrain quantum electrodynamics in Ref. [20].
Of the many features of quantum mechanics and quan-tum field theory (QFT) that dictate the behavior of sub-atomic particles, entanglement and its associated non-locality are perhaps the most striking in their contrastto everyday experience. The degree to which a systemis entangled, or its deviation from tensor-product struc-ture, provides a measure of how “non-classical” it is. Theimportance of entanglement as a feature of quantum the-ory has been known since the work of Einstein, Podolskyand Rosen [21] and later pioneering papers [22–24], andhas become a core ingredient in quantum informationscience, communication and perhaps in understandingthe very fabric of spacetime [25–27]. Despite this longhistory, the implications of entanglement in QFTs, e.g.,Refs. [28–39], and in particular for experimental observ-ables in high-energy and heavy-ion collisions are only nowstarting to be explored [20, 40–49]. Here we study therole of entanglement in low-energy nuclear interactions.In general, a low-energy scattering event can entan-
gle position, spin, and flavor quantum numbers, and itis therefore natural to assign an entanglement power tothe S-matrix for nucleon-nucleon scattering. We chooseto define the entanglement power of the S-matrix in atwo-particle spin space [50, 51], noting that this choiceis not unique and that others will be explored elsewhere[52]. This is determined by the action of the S-matrixon an incoming two-particle tensor product state withrandomly-oriented spins, | ini = R(⌦1)|"i1 ⌦ R(⌦2)|"i2,where R(⌦j) is the rotation operator acting in the jth
spin- 12space, and all other quantum numbers associated
with the states have been suppressed. For low-energyprocesses, this random spin pair projects onto the twostates with total spin S = 0, 1, and associated phaseshifts �0,1, in the 1S0 and 3S1 channels, respectively, withprojections onto higher angular momentum states sup-pressed by powers of the nucleon momenta. The entan-glement power, E , of the S-matrix, S, is defined as
E(S) = 1�Z
d⌦1
4⇡
d⌦2
4⇡Tr1
⇥⇢21
⇤, (1)
where ⇢1 = Tr2 [ ⇢12 ] is the reduced density matrixfor particle 1 of the two-particle density matrix ⇢12 =
arX
iv:1
812.
0313
8v1
[nuc
l-th]
7 D
ec 2
018
D. B. Kaplan BAPTS 3/15/19
Entanglement power in s-wave nucleon-nucleon scattering:
2
FIG. 1. The entanglement power, E(S), of the S-matrix as afunction of p, the center-of-mass nucleon momentum. The 1S0
and 3S1 phase shifts used to calculate E(S) were taken fromfour di↵erent models [53–57] to provide a naıve estimate ofsystematic uncertainties. Data for this figure may be foundin Table II in the supplemental material.
| outih out| with | outi = S| ini. By describing the av-erage action of S to transition a tensor-product state toan entangled state, the entanglement power expresses astate-independent entanglement measure that vanisheswhen | outi remains a tensor product state for any | ini.
Following the analysis of Ref. [20], we consider thespin-space entanglement of two distinguishable particles,the proton and neutron for nf = 2 QCD. Neglecting thesmall tensor-force-induced mixing of the 3S1 channel withthe 3D1 channel, the S-matrix for low-energy scatteringbelow inelastic threshold in these sectors can be decom-posed as
S =1
4
�3ei2�1 + ei2�0
�1 +
1
4
�ei2�1 � ei2�0
�� · �, (2)
where 1 = I2 ⌦ I2 and � · � =3P
↵=1
�↵ ⌦ �↵. It follows
that the entanglement power of S is
E(S) = 1
6sin2 (2(�1 � �0)) , (3)
which vanishes when �1 � �0 = m⇡2for any integer m.
This includes the SU(4) symmetric case �1 = �0 wherethe coe�cient of �·� vanishes. Special fixed points wherethe entanglement power vanishes occur when the phaseshifts both vanish, �1 = �0 = 0, or are both at unitarity,�1 = �0 = ⇡
2, or when �1 = 0, �0 = ⇡
2or �1 = ⇡
2, �0 =
0. The S-matrices at these fixed points with vanishingentanglement power are S = ±1 and ±(1+ � · �)/2 2.
The entanglement power in nature is plotted in Fig. 1as a function of the center-of-mass nucleon momentum,p, up to pion production threshold, making use of the1S0 and 3S1 phase shifts derived from the analyses of
2 The S-matrices at the four fixed points realize a representationof the Klein four-group, Z2 ⌦ Z2.
Refs. [53–56]. The four regions indicated are distin-guished by the role of non-perturbative physics. RegionI shows that entanglement power approaches zero in thelimit p ! 0, as will be the case for any finite range inter-action not at unitarity. At momenta around the scaleof the inverse scattering lengths, region II, poles andresonances of S produce highly-entangling interactions.This non-perturbative structure could be considered asource of ultra-low-momentum entanglement power; ex-perimental evidence for this is expected to be found inthe vanishing modification of np-scattering quantum cor-relations at 19.465(42) MeV where the phase shifts dif-fer by ⇡/2 and |p ", n #i scatters into |p #, n "i. In re-gion IV, where energies are of order the chiral symme-try breaking scale, the entangling interactions of quarkand gluon degrees of freedom become prominent. It isregion III that is the main focus of this paper—awayfrom the far-infrared structure but with nucleons as fun-damental degrees of freedom, the entanglement poweris suppressed. Once relativistic corrections and 3S1-3D1
mixing—parametrically suppressed at low-energy—areincluded in Eq. (19), E(S) is expected to remain sup-pressed but non-zero, indicating that the entanglementsuppression in nature is only partial.Much progress has been made in nuclear physics in re-
cent years by considering low-energy e↵ective field theo-ries (EFTs), constrained by data from nucleon scattering.The �0,1 phase shifts can be computed for energies belowthe pion mass, from the pionless EFT for nucleon-nucleoninteractions. The leading interaction in the e↵ective La-grangian is
Lnf=2
LO= �1
2CS(N
†N)2� 1
2CT
�N†�N
�·�N†�N
�, (4)
where N represents both spin states of the proton andneutron fields. These interactions can be re-expressed ascontact interactions in the 1S0 and 3S1 channels with cou-plings C0 = (CS�3CT ) and C1 = (CS+CT ) respectively,where the two couplings are fit to reproduce the 1S0 and3S1 scattering lengths. The C coe�cients both run withthe renormalization group as described in Ref. [58, 59]with a stable IR fixed point at C = 0, corresponding tofree particles, and a nontrivial, unstable IR fixed pointat C = C? corresponding to a divergent scattering lengthand constant phase shift of � = ⇡/2 (the “unitary” fixedpoint). At the four fixed points (described above), where{C0, C1} take the values 0 or C?, the theory has a con-formal (“Schrodinger”) symmetry; there is also a fixedline of enhanced symmetry at CT = 0, or equivalentlyC0 = C1, where the theory possesses the Wigner SU(4)symmetry, as apparent from the form of Eq. (4) withCT = 0. When fitting to the scattering lengths onefinds CT ⌧ CS ' C?, since scattering lengths are un-naturally large in both channels. Therefore, low-energyQCD has approximate SU(4) symmetry and sits closeto the {C?, C?} conformal fixed point [60]. The emer-
2
FIG. 1. The entanglement power, E(S), of the S-matrix as afunction of p, the center-of-mass nucleon momentum. The 1S0
and 3S1 phase shifts used to calculate E(S) were taken fromfour di↵erent models [53–57] to provide a naıve estimate ofsystematic uncertainties. Data for this figure may be foundin Table II in the supplemental material.
| outih out| with | outi = S| ini. By describing the av-erage action of S to transition a tensor-product state toan entangled state, the entanglement power expresses astate-independent entanglement measure that vanisheswhen | outi remains a tensor product state for any | ini.
Following the analysis of Ref. [20], we consider thespin-space entanglement of two distinguishable particles,the proton and neutron for nf = 2 QCD. Neglecting thesmall tensor-force-induced mixing of the 3S1 channel withthe 3D1 channel, the S-matrix for low-energy scatteringbelow inelastic threshold in these sectors can be decom-posed as
S =1
4
�3ei2�1 + ei2�0
�1 +
1
4
�ei2�1 � ei2�0
�� · �, (2)
where 1 = I2 ⌦ I2 and � · � =3P
↵=1
�↵ ⌦ �↵. It follows
that the entanglement power of S is
E(S) = 1
6sin2 (2(�1 � �0)) , (3)
which vanishes when �1 � �0 = m⇡2for any integer m.
This includes the SU(4) symmetric case �1 = �0 wherethe coe�cient of �·� vanishes. Special fixed points wherethe entanglement power vanishes occur when the phaseshifts both vanish, �1 = �0 = 0, or are both at unitarity,�1 = �0 = ⇡
2, or when �1 = 0, �0 = ⇡
2or �1 = ⇡
2, �0 =
0. The S-matrices at these fixed points with vanishingentanglement power are S = ±1 and ±(1+ � · �)/2 2.
The entanglement power in nature is plotted in Fig. 1as a function of the center-of-mass nucleon momentum,p, up to pion production threshold, making use of the1S0 and 3S1 phase shifts derived from the analyses of
2 The S-matrices at the four fixed points realize a representationof the Klein four-group, Z2 ⌦ Z2.
Refs. [53–56]. The four regions indicated are distin-guished by the role of non-perturbative physics. RegionI shows that entanglement power approaches zero in thelimit p ! 0, as will be the case for any finite range inter-action not at unitarity. At momenta around the scaleof the inverse scattering lengths, region II, poles andresonances of S produce highly-entangling interactions.This non-perturbative structure could be considered asource of ultra-low-momentum entanglement power; ex-perimental evidence for this is expected to be found inthe vanishing modification of np-scattering quantum cor-relations at 19.465(42) MeV where the phase shifts dif-fer by ⇡/2 and |p ", n #i scatters into |p #, n "i. In re-gion IV, where energies are of order the chiral symme-try breaking scale, the entangling interactions of quarkand gluon degrees of freedom become prominent. It isregion III that is the main focus of this paper—awayfrom the far-infrared structure but with nucleons as fun-damental degrees of freedom, the entanglement poweris suppressed. Once relativistic corrections and 3S1-3D1
mixing—parametrically suppressed at low-energy—areincluded in Eq. (19), E(S) is expected to remain sup-pressed but non-zero, indicating that the entanglementsuppression in nature is only partial.Much progress has been made in nuclear physics in re-
cent years by considering low-energy e↵ective field theo-ries (EFTs), constrained by data from nucleon scattering.The �0,1 phase shifts can be computed for energies belowthe pion mass, from the pionless EFT for nucleon-nucleoninteractions. The leading interaction in the e↵ective La-grangian is
Lnf=2
LO= �1
2CS(N
†N)2� 1
2CT
�N†�N
�·�N†�N
�, (4)
where N represents both spin states of the proton andneutron fields. These interactions can be re-expressed ascontact interactions in the 1S0 and 3S1 channels with cou-plings C0 = (CS�3CT ) and C1 = (CS+CT ) respectively,where the two couplings are fit to reproduce the 1S0 and3S1 scattering lengths. The C coe�cients both run withthe renormalization group as described in Ref. [58, 59]with a stable IR fixed point at C = 0, corresponding tofree particles, and a nontrivial, unstable IR fixed pointat C = C? corresponding to a divergent scattering lengthand constant phase shift of � = ⇡/2 (the “unitary” fixedpoint). At the four fixed points (described above), where{C0, C1} take the values 0 or C?, the theory has a con-formal (“Schrodinger”) symmetry; there is also a fixedline of enhanced symmetry at CT = 0, or equivalentlyC0 = C1, where the theory possesses the Wigner SU(4)symmetry, as apparent from the form of Eq. (4) withCT = 0. When fitting to the scattering lengths onefinds CT ⌧ CS ' C?, since scattering lengths are un-naturally large in both channels. Therefore, low-energyQCD has approximate SU(4) symmetry and sits closeto the {C?, C?} conformal fixed point [60]. The emer-
D. B. Kaplan BAPTS 3/15/19
Entanglement power in s-wave nucleon-nucleon scattering:
2
FIG. 1. The entanglement power, E(S), of the S-matrix as afunction of p, the center-of-mass nucleon momentum. The 1S0
and 3S1 phase shifts used to calculate E(S) were taken fromfour di↵erent models [53–57] to provide a naıve estimate ofsystematic uncertainties. Data for this figure may be foundin Table II in the supplemental material.
| outih out| with | outi = S| ini. By describing the av-erage action of S to transition a tensor-product state toan entangled state, the entanglement power expresses astate-independent entanglement measure that vanisheswhen | outi remains a tensor product state for any | ini.
Following the analysis of Ref. [20], we consider thespin-space entanglement of two distinguishable particles,the proton and neutron for nf = 2 QCD. Neglecting thesmall tensor-force-induced mixing of the 3S1 channel withthe 3D1 channel, the S-matrix for low-energy scatteringbelow inelastic threshold in these sectors can be decom-posed as
S =1
4
�3ei2�1 + ei2�0
�1 +
1
4
�ei2�1 � ei2�0
�� · �, (2)
where 1 = I2 ⌦ I2 and � · � =3P
↵=1
�↵ ⌦ �↵. It follows
that the entanglement power of S is
E(S) = 1
6sin2 (2(�1 � �0)) , (3)
which vanishes when �1 � �0 = m⇡2for any integer m.
This includes the SU(4) symmetric case �1 = �0 wherethe coe�cient of �·� vanishes. Special fixed points wherethe entanglement power vanishes occur when the phaseshifts both vanish, �1 = �0 = 0, or are both at unitarity,�1 = �0 = ⇡
2, or when �1 = 0, �0 = ⇡
2or �1 = ⇡
2, �0 =
0. The S-matrices at these fixed points with vanishingentanglement power are S = ±1 and ±(1+ � · �)/2 2.
The entanglement power in nature is plotted in Fig. 1as a function of the center-of-mass nucleon momentum,p, up to pion production threshold, making use of the1S0 and 3S1 phase shifts derived from the analyses of
2 The S-matrices at the four fixed points realize a representationof the Klein four-group, Z2 ⌦ Z2.
Refs. [53–56]. The four regions indicated are distin-guished by the role of non-perturbative physics. RegionI shows that entanglement power approaches zero in thelimit p ! 0, as will be the case for any finite range inter-action not at unitarity. At momenta around the scaleof the inverse scattering lengths, region II, poles andresonances of S produce highly-entangling interactions.This non-perturbative structure could be considered asource of ultra-low-momentum entanglement power; ex-perimental evidence for this is expected to be found inthe vanishing modification of np-scattering quantum cor-relations at 19.465(42) MeV where the phase shifts dif-fer by ⇡/2 and |p ", n #i scatters into |p #, n "i. In re-gion IV, where energies are of order the chiral symme-try breaking scale, the entangling interactions of quarkand gluon degrees of freedom become prominent. It isregion III that is the main focus of this paper—awayfrom the far-infrared structure but with nucleons as fun-damental degrees of freedom, the entanglement poweris suppressed. Once relativistic corrections and 3S1-3D1
mixing—parametrically suppressed at low-energy—areincluded in Eq. (19), E(S) is expected to remain sup-pressed but non-zero, indicating that the entanglementsuppression in nature is only partial.Much progress has been made in nuclear physics in re-
cent years by considering low-energy e↵ective field theo-ries (EFTs), constrained by data from nucleon scattering.The �0,1 phase shifts can be computed for energies belowthe pion mass, from the pionless EFT for nucleon-nucleoninteractions. The leading interaction in the e↵ective La-grangian is
Lnf=2
LO= �1
2CS(N
†N)2� 1
2CT
�N†�N
�·�N†�N
�, (4)
where N represents both spin states of the proton andneutron fields. These interactions can be re-expressed ascontact interactions in the 1S0 and 3S1 channels with cou-plings C0 = (CS�3CT ) and C1 = (CS+CT ) respectively,where the two couplings are fit to reproduce the 1S0 and3S1 scattering lengths. The C coe�cients both run withthe renormalization group as described in Ref. [58, 59]with a stable IR fixed point at C = 0, corresponding tofree particles, and a nontrivial, unstable IR fixed pointat C = C? corresponding to a divergent scattering lengthand constant phase shift of � = ⇡/2 (the “unitary” fixedpoint). At the four fixed points (described above), where{C0, C1} take the values 0 or C?, the theory has a con-formal (“Schrodinger”) symmetry; there is also a fixedline of enhanced symmetry at CT = 0, or equivalentlyC0 = C1, where the theory possesses the Wigner SU(4)symmetry, as apparent from the form of Eq. (4) withCT = 0. When fitting to the scattering lengths onefinds CT ⌧ CS ' C?, since scattering lengths are un-naturally large in both channels. Therefore, low-energyQCD has approximate SU(4) symmetry and sits closeto the {C?, C?} conformal fixed point [60]. The emer-
2
FIG. 1. The entanglement power, E(S), of the S-matrix as afunction of p, the center-of-mass nucleon momentum. The 1S0
and 3S1 phase shifts used to calculate E(S) were taken fromfour di↵erent models [53–57] to provide a naıve estimate ofsystematic uncertainties. Data for this figure may be foundin Table II in the supplemental material.
| outih out| with | outi = S| ini. By describing the av-erage action of S to transition a tensor-product state toan entangled state, the entanglement power expresses astate-independent entanglement measure that vanisheswhen | outi remains a tensor product state for any | ini.
Following the analysis of Ref. [20], we consider thespin-space entanglement of two distinguishable particles,the proton and neutron for nf = 2 QCD. Neglecting thesmall tensor-force-induced mixing of the 3S1 channel withthe 3D1 channel, the S-matrix for low-energy scatteringbelow inelastic threshold in these sectors can be decom-posed as
S =1
4
�3ei2�1 + ei2�0
�1 +
1
4
�ei2�1 � ei2�0
�� · �, (2)
where 1 = I2 ⌦ I2 and � · � =3P
↵=1
�↵ ⌦ �↵. It follows
that the entanglement power of S is
E(S) = 1
6sin2 (2(�1 � �0)) , (3)
which vanishes when �1 � �0 = m⇡2for any integer m.
This includes the SU(4) symmetric case �1 = �0 wherethe coe�cient of �·� vanishes. Special fixed points wherethe entanglement power vanishes occur when the phaseshifts both vanish, �1 = �0 = 0, or are both at unitarity,�1 = �0 = ⇡
2, or when �1 = 0, �0 = ⇡
2or �1 = ⇡
2, �0 =
0. The S-matrices at these fixed points with vanishingentanglement power are S = ±1 and ±(1+ � · �)/2 2.
The entanglement power in nature is plotted in Fig. 1as a function of the center-of-mass nucleon momentum,p, up to pion production threshold, making use of the1S0 and 3S1 phase shifts derived from the analyses of
2 The S-matrices at the four fixed points realize a representationof the Klein four-group, Z2 ⌦ Z2.
Refs. [53–56]. The four regions indicated are distin-guished by the role of non-perturbative physics. RegionI shows that entanglement power approaches zero in thelimit p ! 0, as will be the case for any finite range inter-action not at unitarity. At momenta around the scaleof the inverse scattering lengths, region II, poles andresonances of S produce highly-entangling interactions.This non-perturbative structure could be considered asource of ultra-low-momentum entanglement power; ex-perimental evidence for this is expected to be found inthe vanishing modification of np-scattering quantum cor-relations at 19.465(42) MeV where the phase shifts dif-fer by ⇡/2 and |p ", n #i scatters into |p #, n "i. In re-gion IV, where energies are of order the chiral symme-try breaking scale, the entangling interactions of quarkand gluon degrees of freedom become prominent. It isregion III that is the main focus of this paper—awayfrom the far-infrared structure but with nucleons as fun-damental degrees of freedom, the entanglement poweris suppressed. Once relativistic corrections and 3S1-3D1
mixing—parametrically suppressed at low-energy—areincluded in Eq. (19), E(S) is expected to remain sup-pressed but non-zero, indicating that the entanglementsuppression in nature is only partial.Much progress has been made in nuclear physics in re-
cent years by considering low-energy e↵ective field theo-ries (EFTs), constrained by data from nucleon scattering.The �0,1 phase shifts can be computed for energies belowthe pion mass, from the pionless EFT for nucleon-nucleoninteractions. The leading interaction in the e↵ective La-grangian is
Lnf=2
LO= �1
2CS(N
†N)2� 1
2CT
�N†�N
�·�N†�N
�, (4)
where N represents both spin states of the proton andneutron fields. These interactions can be re-expressed ascontact interactions in the 1S0 and 3S1 channels with cou-plings C0 = (CS�3CT ) and C1 = (CS+CT ) respectively,where the two couplings are fit to reproduce the 1S0 and3S1 scattering lengths. The C coe�cients both run withthe renormalization group as described in Ref. [58, 59]with a stable IR fixed point at C = 0, corresponding tofree particles, and a nontrivial, unstable IR fixed pointat C = C? corresponding to a divergent scattering lengthand constant phase shift of � = ⇡/2 (the “unitary” fixedpoint). At the four fixed points (described above), where{C0, C1} take the values 0 or C?, the theory has a con-formal (“Schrodinger”) symmetry; there is also a fixedline of enhanced symmetry at CT = 0, or equivalentlyC0 = C1, where the theory possesses the Wigner SU(4)symmetry, as apparent from the form of Eq. (4) withCT = 0. When fitting to the scattering lengths onefinds CT ⌧ CS ' C?, since scattering lengths are un-naturally large in both channels. Therefore, low-energyQCD has approximate SU(4) symmetry and sits closeto the {C?, C?} conformal fixed point [60]. The emer-
Use models which fit the phase shift data accurately:2
FIG. 1. The entanglement power, E(S), of the S-matrix as afunction of p, the center-of-mass nucleon momentum. The 1S0
and 3S1 phase shifts used to calculate E(S) were taken fromfour di↵erent models [53–57] to provide a naıve estimate ofsystematic uncertainties. Data for this figure may be foundin Table II in the supplemental material.
| outih out| with | outi = S| ini. By describing the av-erage action of S to transition a tensor-product state toan entangled state, the entanglement power expresses astate-independent entanglement measure that vanisheswhen | outi remains a tensor product state for any | ini.
Following the analysis of Ref. [20], we consider thespin-space entanglement of two distinguishable particles,the proton and neutron for nf = 2 QCD. Neglecting thesmall tensor-force-induced mixing of the 3S1 channel withthe 3D1 channel, the S-matrix for low-energy scatteringbelow inelastic threshold in these sectors can be decom-posed as
S =1
4
�3ei2�1 + ei2�0
�1 +
1
4
�ei2�1 � ei2�0
�� · �, (2)
where 1 = I2 ⌦ I2 and � · � =3P
↵=1
�↵ ⌦ �↵. It follows
that the entanglement power of S is
E(S) = 1
6sin2 (2(�1 � �0)) , (3)
which vanishes when �1 � �0 = m⇡2for any integer m.
This includes the SU(4) symmetric case �1 = �0 wherethe coe�cient of �·� vanishes. Special fixed points wherethe entanglement power vanishes occur when the phaseshifts both vanish, �1 = �0 = 0, or are both at unitarity,�1 = �0 = ⇡
2, or when �1 = 0, �0 = ⇡
2or �1 = ⇡
2, �0 =
0. The S-matrices at these fixed points with vanishingentanglement power are S = ±1 and ±(1+ � · �)/2 2.
The entanglement power in nature is plotted in Fig. 1as a function of the center-of-mass nucleon momentum,p, up to pion production threshold, making use of the1S0 and 3S1 phase shifts derived from the analyses of
2 The S-matrices at the four fixed points realize a representationof the Klein four-group, Z2 ⌦ Z2.
Refs. [53–56]. The four regions indicated are distin-guished by the role of non-perturbative physics. RegionI shows that entanglement power approaches zero in thelimit p ! 0, as will be the case for any finite range inter-action not at unitarity. At momenta around the scaleof the inverse scattering lengths, region II, poles andresonances of S produce highly-entangling interactions.This non-perturbative structure could be considered asource of ultra-low-momentum entanglement power; ex-perimental evidence for this is expected to be found inthe vanishing modification of np-scattering quantum cor-relations at 19.465(42) MeV where the phase shifts dif-fer by ⇡/2 and |p ", n #i scatters into |p #, n "i. In re-gion IV, where energies are of order the chiral symme-try breaking scale, the entangling interactions of quarkand gluon degrees of freedom become prominent. It isregion III that is the main focus of this paper—awayfrom the far-infrared structure but with nucleons as fun-damental degrees of freedom, the entanglement poweris suppressed. Once relativistic corrections and 3S1-3D1
mixing—parametrically suppressed at low-energy—areincluded in Eq. (19), E(S) is expected to remain sup-pressed but non-zero, indicating that the entanglementsuppression in nature is only partial.Much progress has been made in nuclear physics in re-
cent years by considering low-energy e↵ective field theo-ries (EFTs), constrained by data from nucleon scattering.The �0,1 phase shifts can be computed for energies belowthe pion mass, from the pionless EFT for nucleon-nucleoninteractions. The leading interaction in the e↵ective La-grangian is
Lnf=2
LO= �1
2CS(N
†N)2� 1
2CT
�N†�N
�·�N†�N
�, (4)
where N represents both spin states of the proton andneutron fields. These interactions can be re-expressed ascontact interactions in the 1S0 and 3S1 channels with cou-plings C0 = (CS�3CT ) and C1 = (CS+CT ) respectively,where the two couplings are fit to reproduce the 1S0 and3S1 scattering lengths. The C coe�cients both run withthe renormalization group as described in Ref. [58, 59]with a stable IR fixed point at C = 0, corresponding tofree particles, and a nontrivial, unstable IR fixed pointat C = C? corresponding to a divergent scattering lengthand constant phase shift of � = ⇡/2 (the “unitary” fixedpoint). At the four fixed points (described above), where{C0, C1} take the values 0 or C?, the theory has a con-formal (“Schrodinger”) symmetry; there is also a fixedline of enhanced symmetry at CT = 0, or equivalentlyC0 = C1, where the theory possesses the Wigner SU(4)symmetry, as apparent from the form of Eq. (4) withCT = 0. When fitting to the scattering lengths onefinds CT ⌧ CS ' C?, since scattering lengths are un-naturally large in both channels. Therefore, low-energyQCD has approximate SU(4) symmetry and sits closeto the {C?, C?} conformal fixed point [60]. The emer-
D. B. Kaplan BAPTS 3/15/19
1. Implications of spin-flavor symmetry in effective nuclear forces
Short distance nuclear forces relevant for low energy processes can be incorporated
into chiral Lagrangians in terms of local operators in a derivative expansion [1,2]. There
are two leading (dimension six) operators involving nucleons alone, given by
L6 = −12CS(N †N)2 − 1
2CT (N †σN)2 (1.1)
where N are isodoublet two-component spinors, and the σ are Pauli matrices. Higher
derivative operators account for the spin-orbit coupling, among other effects1. Including
the ∆ isobars in the theory leads to 18 independent dimension six operators allowed by spin
and isospin symmetry2. In order to discuss hypernuclei, or strangeness in dense matter,
one must consider SU(3) flavor multiplets — there are six independent leading operators
involving the baryon octet alone [3], while including the decuplet inflates the number to 28
independent operators. The number of independent dimension seven interactions is still
much greater.
Clearly, to make headway in a systematic effective field theory analysis of nuclear and
hypernuclear forces, it is desirable to find some simplifying principle. In this letter we
propose that among the baryon interactions, SU(4) spin-flavor symmetry for two flavors,
or SU(6) symmetry for three flavors should be a good approximation. We show how these
symmetries have a vastly simplifying effect on the dimension six interactions described
above, reducing both the 18 N−∆ interactions and the 28 octet-decuplet interactions down
to just two independent operators. We support our allegation that spin-flavor symmetry is
relevant to nuclear forces first by outlining its implications and by giving empirical evidence
in support of SU(4) in nuclei. Then we prove that these symmetries become exact in the
large-N limit of QCD.
1 In low energy nucleon-nucleon scattering the higher derivative terms will be less important
than the leading operator. However, many-body effects in large nuclei can enhance the importance
of subleading operators, such as the spin-orbit interaction.2 It is simplest to count operators in the form (ψ1ψ2)(ψ3ψ4)
†, requiring (ψ1ψ2) and (ψ3ψ4) to
have the same spin and isospin quantum numbers. One finds the above two (NN)(NN)† operators;
zero operators of the form (NN)(N∆)†; two (NN)(∆∆)†, four (N∆)(N∆)†, two (N∆)(∆∆)†,
and eight (∆∆)(∆∆)† operators.
1
1S0 : C0 = (CS � 3CT )3S1 : C1 = (CS + CT )
Alternatively: look at the space of low energy EFTs for pcm < mπ/2
Conformal fixed points (infinite scattering lengths)
SU(4)Wigner symmetry line
D. B. Kaplan BAPTS 3/15/19
Other definitions of entanglement power also explored
Can show in each case:
• SU(4)Wigner for Nf=2
• SU(16) for Nf=3
• Conformal symmetry, for Nf=2,3
Are sufficient to ensure zero entanglement power for the S-matrix
… and probably necessary.
D. B. Kaplan BAPTS 3/15/19
Conclusions:
There are apparent approximate symmetries w/o explanation in the strong interactions:
• non-quark spin-flavor symmetries • NR conformal (Schrödinger) symmetries
Can ascribe an “entanglement power” to the S-matrix which knows about flavor & spin changing interactions
Entanglement is minimized for flavors & spin diagonal interactions, as well as for conformal fixed points
Can symmetries be explained by dynamical systems “wanting” to minimize entanglement?
Need to examine more examples; model examples with feedback mechanism